Pat the salmon fillets dry using paper towels to remove excess moisture. Season both sides with salt and pepper, then lightly dredge in flour. This thin coating helps create a golden crust while keeping the fish tender.
Heat olive oil in a large skillet over medium heat. Place the salmon skin-side down and cook for about 4 minutes per side until golden and just cooked through. Remove the salmon from the pan and set it aside.
In the same skillet, melt butter and add minced garlic. Cook for about 30 seconds until fragrant, being careful not to burn it. This step builds the base flavor for the sauce.
Pour in chicken broth and scrape up any browned bits from the pan. Stir in heavy cream, lemon juice, and lemon zest, letting it simmer gently until slightly thickened.
Mix in the capers and freshly grated Parmesan cheese. Stir until the cheese melts smoothly into the sauce, creating a rich and creamy consistency.
Place the salmon fillets back into the skillet, spooning sauce over them. Let everything simmer for 2–3 minutes so the salmon absorbs the flavors.
Sprinkle fresh parsley over the dish and remove from heat. Serve immediately while the sauce is warm and velvety.